Methods, systems, and computer readable media for improved digital holography and display incorporating same

ABSTRACT

A method for digital holography includes modeling a hologram using a forward propagation model that models propagation of a light field from a hologram plane to an image plane. The method further includes computing the hologram as a solution to an optimization problem that is based on the model. The method further includes configuring at least one spatial light modulator using the hologram. The method further includes illuminating the spatial light modulator using a light source to create a target image.

RELATED APPLICATIONS

This application claims the benefit of U.S. Provisional PatentApplication Ser. No. 62/778,253, filed Dec. 11, 2018; the disclosure ofwhich is incorporated herein by reference in its entirety.

GOVERNMENT INTEREST

This invention was made with government support under Grant NumberCNS-1405847 awarded by the National Science Foundation. The governmenthas certain rights in the invention.

TECHNICAL FIELD

The subject matter described herein relates to digital holography withimproved resolution and reduced visual artifacts. More particularly, thesubject matter described herein relates to methods, systems, andcomputer readable media for improved digital holography and a displayincorporating same.

BACKGROUND

Holography is the process of recording light scattered by an object ontoa holographic recording medium, such as a photographic plate. Once thehologram is recorded on the plate, the original image of the scene canbe reproduced by illuminating the photographic plate with a light sourcethat corresponds to the location of the reference light source usedduring the holographic recording process.

In digital holography, the hologram can be computationally generatedbased on a desired target holographic image and location of the deviceor devices on which the hologram is recorded. Spatial light modulators(SLMs) are commonly used to record the digitally computed hologram. Togenerate the desired holographic image, the spatial light modulator isilluminated with a light source.

Existing methods for computing the hologram used in digital holographyare computationally intensive and introduce visual artifacts into theresulting images. Accordingly, there exists a need for improved methodsand systems for digital holography and a display incorporating improvedmethods and systems.

SUMMARY

A method for digital holography includes modeling a hologram using aforward propagation model that models propagation of a light filed froma hologram plane to an image plane. The method further includescomputing the hologram as a solution to an optimization problem that isbased on the model. The method further includes configuring at least onespatial light modulator using the hologram. The method further includesilluminating the spatial light modulator using a light source to createa target image.

According to one aspect of the subject matter described herein, modelingthe hologram using a forward propagation model includes modeling thehologram using an angular spectrum, Fourier, Fraunhofer, or Fresnelpropagation model.

According to yet another aspect of the subject matter described herein,modeling the hologram using a forward propagation model includesmodeling the hologram as a spatially invariant convolution of thehologram and a lens phase function.

According to yet another aspect of the subject matter described herein,the optimization problem uses a least squares optimization function.

According to yet another aspect of the subject matter described herein,the least squares optimization function comprises:

$\begin{matrix}{minimize} \\H\end{matrix}{{{{H*L}}^{2} - I}}_{2}^{2}$

where H is the hologram, L is the lens phase function, and I is thesquare of the target image intensity.

According to yet another aspect of the subject matter described herein,the hologram comprises a phase hologram for a phase only spatial lightmodulator and the function

$\begin{matrix}{minimize} \\H\end{matrix}{{{{H*L}}^{2} - I}}_{2}^{2}$is approximated using the function:

${\begin{matrix}{\arg\;\min} \\\phi\end{matrix}{{{{H*L}}^{2} - I}}_{2}^{2}},$where ϕ is the phase angle of the hologram for a given object point.

According to yet another aspect of the subject matter described herein,the function

$\begin{matrix}{\arg\;\min} \\\phi\end{matrix}{{{{H*L}}^{2} - I}}_{2}^{2}$is approximated using the following function:

${\begin{matrix}{\arg\;\min} \\\phi\end{matrix}{{{{{F^{H}({FH})}({FL})}}^{2} - I}}_{2}^{2}},$where F is the Fourier transform operator and F^(H) is the inverseFourier transform operator.

According to yet another aspect of the subject matter described herein,computing the hologram includes calculating a gradient of the function

$\begin{matrix}{\arg\;\min} \\\phi\end{matrix}{{{{{{F^{H}({FH})}({FL})}}^{2} - I}}_{2}^{2}.}$

According to yet another aspect of the subject matter described herein,configuring at least one spatial light modulator includes configuring asingle phase only spatial light modulator.

According to yet another aspect of the subject matter described herein,configuring at least one spatial light modulator includes configuringeach of a pair of cascaded phase only spatial light modulators.

According to yet another aspect of the subject matter described herein,computing the hologram includes computing an error Err between areconstructed image |z|² produced by a hologram H and a target image Ifor a penalty function ƒ using the following expression:Err=ƒ(|z| ², I),where z is the complex wave field at the image plane, is a function ofthe hologram H, is the output of the propagation model, and is computedusing the following expression:z=P(H),where P is the propagation model.

According to yet another aspect of the subject matter described herein,the hologram comprises a phase only hologram and the optimizationproblem comprises

${\Phi_{opt} = {{\underset{\Phi}{minimize}\underset{{Err}{(\Phi)}}{\underset{︸}{f( {{{\mathcal{P}( {H(\Phi)} )}}^{2},I} )}}} + {\gamma{{\nabla\Phi}}^{2}}}},$where ∥∇ϕ∥² is a regularizer on phase patterns of the hologram H, γ is aconstant, and Φ is the phase angle of the hologram H.

According to yet another aspect of the subject matter described herein,computing hologram as a solution to the optimization problem comprisesapproximating a gradient of the penalty function ƒ using Wirtingerderivatives.

According to yet another aspect of the subject matter described herein,the penalty function is based on perceptual accuracy or a learnedperceptual similarity metric.

According to yet another aspect of the subject matter described herein,a system for digital holography is provided. The system includes adisplay device. The display device includes at least one processor. Thedisplay device includes a hologram calculator implemented by the atleast one processor and configured for modeling a hologram using aforward propagation model that models propagation of a light field froma hologram plane to an image plane and computing the hologram as asolution to an optimization problem that is based on the model. Thedisplay device further includes at least one spatial light modulator.The display device further includes a spatial light modulator controllerfor configuring the at least one spatial light modulator using thehologram. The display device further includes a light source forilluminating the at least one spatial light modulator to create a targetimage.

The subject matter described herein may be implemented in hardware,software, firmware, or any combination thereof. As such, the terms“function” “node” or “module” as used herein refer to hardware, whichmay also include software and/or firmware components, for implementingthe feature being described. In one exemplary implementation, thesubject matter described herein may be implemented using a computerreadable medium having stored thereon computer executable instructionsthat when executed by the processor of a computer control the computerto perform steps. Exemplary computer readable media suitable forimplementing the subject matter described herein include non-transitorycomputer-readable media, such as disk memory devices, chip memorydevices, programmable logic devices, and application specific integratedcircuits. In addition, a computer readable medium that implements thesubject matter described herein may be located on a single device orcomputing platform or may be distributed across multiple devices orcomputing platforms.

BRIEF DESCRIPTION OF THE DRAWINGS

The patent or application file contains at least one drawing executed incolor. Copies of this patent or patent application publication withcolor drawing(s) will be provided by the Office upon request and paymentof the necessary fee.

The subject matter described herein will now be explained with referenceto the accompanying drawings of which:

FIG. 1 illustrates in the first column, high resolution referenceimages. The second column illustrates images generating using a singleSLM that implements double phase encoding. The third column illustratessimulated images generated using a single SLM whose hologram is computedusing the methods described herein. The fourth column illustratessimulated images generated using cascaded low resolution SLMs whosehologram function is computed using the subject matter described herein.The fifth and sixth columns illustrate magnified view of the images inthe corresponding colored rectangles in the same row.

FIG. 2 is a table comparing the peak signal to noise ratio (PSNR)measurements of holographic images produced using holograms computed bythe methods described herein and holograms from the double phaseencoding method;

FIG. 3 illustrates color artifacts generated using different hologramcomputation methods. In FIG. 3, image A is a reference image. Image B isa reconstruction of the reference image using a single SLM whoseholograph function was generated using the methods described herein.Image C is a reconstruction of the reference image generated using asingle SLM whose hologram function is generated using double phaseencoding. Image D is a reconstruction of the reference image generatedusing cascaded SLMs whose hologram function is computed using themethods described herein.

FIG. 4 is a block diagram of a head-mountable display using an SLMhaving a hologram computed using the methodology described herein;

FIG. 5 is a flow chart illustrating a method for improved digitalholography;

FIG. 6 illustrates Wirtinger holography on a display prototype. Tocompute phase-only modulation patterns, we depart from existingiterative projection algorithms, such as error-reductionGerchberg-Saxton methods [Gerchberg 1972; Peng et al. 2017], andheuristic encoding approximations, such as the double phase encodingmethod [Hsueh and Sawchuk 1978]. Instead, we revisit the use of formaloptimization using complex Wirtinger derivatives for the underlyingphase retrieval problem. We built a near-eye display prototype using aphase-only spatial light modulator (SLM) and off-the-shelf optics(left). Compared to holographic reconstructions at a set focal distancefrom the existing methods [Peng et al. 2017] (center left) and [Maimoneet al. 2017] (center right), the proposed Wirtinger holographysubstantially reduces reconstruction artifacts on our prototype, whileachieving an order of magnitude reduced error in simulation. Therightmost three images of holographic encodings of a mouse embryo imageby Miltenyi Biotec.

FIG. 7 is a schematic diagram of a scalar diffraction model, in whichthe incident wave (Ei) first gets modulated by the phase delays on thehologram, where each pixel then emits a sub-wave (Ep) that propagates infree space to the image plane. The complex amplitude of one point on theimage plane (EO) is the integration of sub-waves propagated from allpixels on the hologram plane. Note, r here represents the Euclideandistance. As such, given a target propagated distance and a principlewavelength, a sharp point spread function on the image plane can beobtained under the point-source propagation model, corresponding to thephase distribution of a lens on the hologram plane.

FIG. 8 (top) is a schematic diagram of our prototype holographicnear-eye display. FIG. 8 (bottom) is an image of the prototypeholographic near-eye display. In the prototype, we couple lasers ofthree different wavelengths into a single-mode fiber whose output endcan be treated as a point source. The light emitted from the single-modefiber is polarized and collimated before incident on the phase-onlyreflective SLM, where the holograms are displayed. The modulated wave isimaged using a camera at the eye position.

FIG. 9 includes graphs illustrating an evaluation of reconstructedholographic images for three different datasets. The GS-based phaseretrieval and our algorithm ran until the PSNR values were converged.The evaluation is done on grayscale images for a 532 nm green lasersource.

FIG. 10 includes images illustrating synthetic results reconstructedfrom holograms generated using different algorithms. From left to right,we show the reference, reconstructed results of modified GS iterativealgorithm, double phase amplitude encoding direct propagation method,and the proposed Wirtinger holography, respectively. Note, for thiscomparison we set the propagated distance 200 mm, hologram pixel pitch6.4 μm, and three principal wavelengths 638 nm, 520 nm, 450 nm forassessment. The reconstructed images are a Christmas motif by Couleur,Rally by Dimitris Vestsikas, Motorbike by Steve Sewell, and Earth byWikilmages.

FIG. 11 illustrates quantitative and qualitative assessments of applyingdifferent penalty functions for the proposed Wirtinger holographymethod.

FIG. 12 illustrates a schematic of a setup for realizing a cascaded SLMholographic display (left), where two SLMs are relayed by optics toperform the phase modulation defined by complex matrix multiplication.The two-phase hologram superresolves a high-resolution phase hologramthat corresponds to reconstruct a high-quality image (center). Syntheticcomparison results produced by different solutions are shown (right).The images that were holographically encoded were obtained fromSandid/Pixabay.

FIG. 13 illustrates experimental results for existing holographic phaseretrieval methods on our prototype display. We show here single-colorreconstructions for the green and red wavelength channels of our system.The phase-only holograms used in this prototype are provided in thesection entitled Supplementary Information for Wirtinger Holography forNear-Eye Displays. The images in each row are captured with same camerasettings, using ISO 100 and the same exposure time of 10 ms.

FIG. 14 illustrates experimental results for existing holographic phaseretrieval methods on our prototype display. We present RGB color imageswith each color channel captured sequentially. The phase-only hologramsused in this prototype are provided in the section below entitledSupplementary Information for Wirtinger Holography for Near-EyeDisplays. The images in each row are captured with the same camerasettings, using ISO 100 and an exposure time of 10 ms. We have tuned theoutput power of three lasers before acquisition to approximatelywhite-balance the illumination. The actin cell image was captured by JanSchmoranzer.

FIG. 15 illustrates complications in computing a high-quality 3Dhologram. It is difficult to simultaneously create closely spaced brightand dark voxels in space with high fidelity. Creating a dark voxelimmediately after a bright voxel, for any 3D hologram, requires rapidchange in phase and is infeasible for a free space propagation.

FIG. 16 illustrates simulated holograms for a 3D scene generated usingfast depth switching and dynamic global scene focus for an assumed eyetracked position. Observe that the depth of field effects areappropriately represented without compromising on the holographic imagequality. The depth of the above scene is scaled to make the neardistance at 200 mm and far distance at 500 mm. The images are from theLytro dataset.

FIG. 17 illustrates an alternative optical design of the prototypenear-eye display. Note that the laser source and its correspondingcollimating optics are not shown in FIG. 17.

FIG. 18 illustrates PSNR values plotted against number of iterations forWirtinger holography optimization for a randomly initialized phase.

FIG. 19 illustrates the probability of detection of errors for thesimulated image also shown in FIG. 10. Note that the Wirtinger hologramsproduce much less perceptual error as compared to that of double phaseencoding method.

FIG. 20 illustrates a comparison of holographic images from a cascade oftwo low-resolution SLMs with hologram reconstructions of bothdouble-phase and Wirtinger holograms from a high-resolution SLM. Notethat the cascaded low-resolution holograms are significantly better inquality compared to high resolution double phase holograms.

FIG. 21 illustrates zero order diffracted light from the SLM for allthree red, green and blue color lasers.

FIG. 22 illustrates a simulation of zero order filtering: Filtering theZero order light improves the holographic image quality significantly.

FIG. 23 illustrates phase patterns of green channel for the imagespresented in FIG. 13.

FIG. 24 illustrates results reconstructed by modified GS holograms,Double phase encoding of Fresnel holograms, and Wirtinger Holograms. Theneurons image was captured by Karthik Krishnamurthy © MBF Bioscience.

FIG. 25 illustrates Additional comparison of single-color resultsgenerated by double phase encoding (left) and Wirtinger Holograms(right).

DETAILED DESCRIPTION

This disclosure is organized into three main sections. The first sectionis entitled Hogel-Free Holography. The second section is entitledWirtinger Holography. The third section is entitled SupplementaryInformation for Wirtinger Holography for Near Eye Displays.

Hogel-Free Holography

1 Overview

1.1 Modeling the Hologram

Our forward model for generating the hologram is based on the principleof Fresnel holography. Light propagating from a source is modulated by aspatial light modulator (SLM) device in what is called a hologram plane.When the light encounters the hologram (or SLM) plane, it is diffractedto focus at various encoded object points thus forming a volumetricimage. Such phase alterations on the SLM that result in formation ofobject points by phase modulated light is referred to as lens phasefunction. A lens phase function can be thought of as a physical lensthat is encoded on an SLM. The spatial frequencies encoded in such aholographic lens determine the deflection of the incident light, whichhigher spatial frequencies deflecting the light more. Since the SLMsupports only limited range of frequencies, the lens phase functionscorresponding to each point is confined to a local region on the SLM.Mathematically, such a lens phase function representing an object pointo can be defined as

${L( \overset{arrow}{p} )}_{o} = e^{i(\frac{2\pi{{\overset{arrow}{p} - \overset{arrow}{o}}}}{\lambda})}$where {right arrow over (p)} is the pixel on the local confinement ofthe lens phase function on the SLM and λ is the wavelength of light.

An additive superposition of several such lens phase functions, eachrepresenting a unique object point, weighted by their respective imageintensities form a complex hologram. This can be represented asmathematically asH({right arrow over (p)})=Σ_(j)A_(j)L({right arrow over (p)})_(j)where j runs over set of all object points which have contributions fromthe pixel p on the SLM, and the intensity of object point at {rightarrow over (j)} is A_(j) ².

Note that the hologram is an explicit integration of all object pointsin the scene. However, this is equivalent to the convolution of thetarget image with spatially varying complex valued lens phase kernel. Ifthe spatial variance is removed, so that the lens phase function is samefor all object points, the calculation of such a hologram can be sped upby replacing the point-wise integration with a spatially invariantconvolution:H=A*Lwhere * is the convolution operator, A is the square root of targetimage intensities and L is the spatially invariant lens phase kernel.Similarly, given a hologram H, the underlying image can be obtained byinverting the process, i.e. deconvolving the Hologram with the lenskernel. It can be observed that the lens phase kernel is a unitarymatrix and the deconvolution of hologram can be thought of asconvolution with the conjugate lens phase kernel.

Hence the hologram corresponding to a given target image can be thoughtof as a solution to the following least squares optimization problem.

$\begin{matrix}{minimize} \\H\end{matrix}{{{{H*L}}^{2} - I}}_{2}^{2}$Note that |.|² is an element-wise square, H is the hologram, L is thecomplex lens phase function and I is the target image. Note that thematrices in the above equation are all vectorized, and hereaftervectorized matrices and normal matrices might be used interchangeablyfor convenience of representation.

More precisely, since we want to compute the phase hologram for aphase-only SLM, the above optimization is equivalent to

$\underset{\phi}{\arg\;\min}{{{{H*L}}^{2} - I}}_{2}^{2}$where H=exp(jϕ) is a phase-only hologram with unit amplitude.

Since convolution is equivalent to multiplication in Fourier domain,

$\underset{\phi}{\arg\;\min}{{{{{F^{H}({FH})}({FL})}}^{2} - I}}_{2}^{2}$where F is a Fourier transform operator and F^(H) is the hermitian of itresulting in inverse Fourier transform.

Define z=F^(H)(FH)(FL) and ƒ(z)=∥|z|²−I∥₂ ², then the loss functionE(H)=∥|F^(H)(FH)(FL)|²−I∥₂ ² is equivalent to

E(H) = f(z) $\begin{matrix}{{d( {E(H)} )} = {{df}(z)}} \\{= {{Re}\langle {{\nabla f},{dz}} \rangle}} \\{= {{Re}\langle {{\nabla f},{d( {{F^{H}({FH})}({FL})} )}} \rangle}} \\{= {{Re}\langle {{\nabla f},{F^{H}{FLFdH}}} \rangle}} \\{= {{Re}\langle {{{F^{H}({FL})}^{H}F{\nabla f}},{dH}} \rangle}} \\{= {{Re}\langle {{{F^{H}({FL})}^{*}F{\nabla f}},{dH}} \rangle}} \\{= {{Re}\langle {{{F^{H}({FL})}^{*}F{\nabla f}},{d( e^{j\;\phi} )}} \rangle}} \\{= {{Re}\langle {{{je}^{j\;\phi}{F^{H}({FL})}^{*}F{\nabla f}},{d(\phi)}} \rangle}}\end{matrix}$Note that L is a unitary and symmetric matrix in our case, and so is FL.And hence the hermitian is equal to the complex conjugate.

From the definition of ƒ(z), we can obtain the gradient of ƒ as

$\begin{matrix}{{\nabla f} = {2{\nabla_{z}f}}} \\{= {2{\nabla_{z}( {{{z}^{2} - I}} )}}} \\{= {2{( {{z}^{2} - I} ) \circ {\nabla_{z}( {z}^{2} )}}}} \\{= {2{( {{z}^{2} - I} ) \circ {\nabla_{z}( {z\overset{\_}{z}} )}}}} \\{= {2{( {{z}^{2} - I} ) \circ z}}}\end{matrix}$where the operator ∘ denotes the Hadamard product or element-wiseproduct.

With the above objective function and the gradient, the nonconvexproblem can be solved with quasi-newton method or interior-point methodwith L-BFGS updates for images of large sizes.

For the cascaded SLM version, the complete hologram is contributed byboth SLMs in a multiplicative fashion for the field, or additive inphase. Therefore the hologram H can be defined asH=H₁ ∘H ₂where H1=exp(jϕ₁) and H2=exp(iϕ₂) are solved for in an alternating leastsquares sense.2 Evaluation and AnalysisNOTE: Double phase encoding with object phase compensation as employedby [Maimone et al. 2017] is referred to as just double phase encodinghere for simplicity.

In this section we quantitatively and qualitatively analyze ouralgorithm for generating holograms, both for a single phase-only SLM andtwo cascaded SLMs for super resolution, both in simulation and byexperiment.

2.1 Software Simulation

We evaluate the theoretical performance and achieved quality of theholograms generated by our method, in simulation, by numericallyreconstructing the holographic image. For this, holograms for variousimages are generated for an SLM of resolution 1920×1080 pixels, with alens phase kernel of size 176 pixels along each dimension. All fullcolor holograms are generated in a color sequential manner, simulatingthe physical prototype.

The numerical reconstruction was done as an inverse process of ourhologram generation forward model, i.e. by deconvolving the hologramwith the lens phase function. We also simulate the superresolvedholographic images generated by cascading two SLMs each of half theresolution, i.e. 960×540 pixels. All the input images used for computingthe holograms as well as the holographic output are maintained to beequal to 1920×1080 in resolution. Note that our hologram generationforward model is partially similar to that of double phase encodinghologram (DPH) which produced noise-free and high-qualityreconstructions [Maimone et al. 2017]. Therefore, for each example, wecompare our approach with DPH by keeping the settings of hologramgeneration process identical in both cases.

2.1.1 Quantitative Analysis.

FIG. 1 compares image patches reconstructed from holograms computed byour method, against those computed by double phase encoding algorithm.The table illustrated in FIG. 2 provides quantitative comparisons of theimages using PSNR measurements. The intensity of the reconstructed imageis adjusted to have the same mean value of the corresponding originalimage for analyzing the holographic image quality of images produced byall methods [Yoshikawa et al. 2016].

Several observations can be made from visual comparisons and the qualitymetric table. Foremost, we find that our method outperforms double phaseencoding holograms for a single SLM by a significant margin. Moreover,for superresolved images from cascaded low resolution SLMs, our methodoutperform double phase encoding, resulting in superior qualityreconstructed images, and closely approach holographic images generatedby our method from a single high-resolution phase SLM. We emphasize thatthe cascaded SLM implementation spatially superresolved images by afactor of four, i.e. twice as many pixels along each axis. Apart fromthese metrics which provide a global evaluation of the image quality, wealso provide magnified views in FIG. 1 which we find to give morevisually meaningful information to the high error region.

The advantages gained in the PSNR values directly translate to visiblereductions in artifacts: a detailed qualitative analysis is done in thenext section. These measurements highlight the performance of ouralgorithm and the promise of using cascaded SLMs for producing superiorquality holographic images of quadrupled resolution, a capability thathas not been achieved before to the best of our knowledge.

2.1.2 Qualitative Analysis

We notice three major artifacts that substantially degrade the qualityof holographic images produced by double phase encoding method: 1)resolution loss due to direct amplitude encoding, 2) ringing artifactsand 3) color artifacts. In this section, we qualitatively analyze theperformance our algorithm. FIG. 1 compares image patches produced by ourmethod, both for a single SLM and a cascade of two low resolution SLMs,against double phase holograms.

Resolution

Although the double phase encoding method adopted by Maimone et al.[2017] produce a close approximation to the target image, encoding asingle complex value into two phase values result in loss of resolutionof holographic images. It can be observed that our method produces highquality reconstructed images without any effective loss of resolution.Moreover, our cascaded hologram implementation successfully producedimages effectively of resolution four times the resolution of any singleSLM. For example, notice the enhancement of text, in FIG. 1, and thescales on the ruler.

Ringing

The ringing effect, or Gibbs phenomenon, is a rippling artifact nearsharp edges caused by either distortion or loss of high frequencyinformation. Although ringing artifacts arising from double phaseencoding are subtle for smooth images, they are very prominent forimages with many high frequency details. Images in the last two rows inFIG. 1 shows an example of reconstruction of a newspaper and a ruler. Itcan be observed how the ringing artifacts degrade the reconstructedimage quality in double phase hologram. Our method produces hologramswith no such ringing artifacts. The cascaded hologram computed by ourmethod also produces faithful reconstruction of the target images,sometimes with subtle noise, which is not visible unless zoomedsufficiently.

Color Artifacts

We observed that sometimes the reconstructed holographic image from adouble phase hologram produce color artifacts, mainly on patches ofwhite or gray. Our method produced accurate numerical reconstructionsfor both single SLM hologram as well as cascaded hologram. FIG. 3 showsa scenario where our method performs significantly better than doublephase encoding method. The undesired green colorations can be observedon the holographic image produced by a double phase hologram whereas ourmethod produced accurate reconstructions, both for single and cascadedSLMs.

FIG. 4 is a block diagram of a head-mountable display that incorporatesa spatial light modulator that implements the holograms computed usingthe methodology described herein. The display illustrated in FIG. 4 mayimplement Hogel-free holography as described in this section orWirtinger holography as described in the sections below. Referring toFIG. 4, a head-mountable display 100 includes a form factor that isdesigned to be worn on a user's head. For example, head-mountabledisplay 100 may include a frame that has a form factor designed to beworn on a user's head. For example, the frame may have an eyeglasses orhelmet-like form factor where a display screen is placed in front of auser's eyes. Head-mountable display 100 may be an AR display where theuser views synthetic images and real-world images. Alternatively,head-mountable display may be a VR display where the user views onlysynthetic images.

In the illustrated example, head-mountable display 100 includes aprocessor 102 that implements a hologram calculation function 104.Processor 102 may be any suitable processor, such as a graphicsprocessing unit. Hologram calculation function 104 calculates hologramsusing the methodology described herein to generate target holographicimages. Under control of hologram calculation function 104, processor102 instructs a spatial light modulator controller 106 to configure aspatial light modulator 108 to generate the computed hologram functions.Spatial light modulator 108 may be a configurable device capable ofimplementing diffraction patterns corresponding to the calculatedhologram functions. As stated above, spatial light modulators may becascaded to implement a hologram. A light source 110 illuminates spatiallight modulator 108 to generate desired or target holographic images.

Although FIG. 4 illustrates a head mountable display, such as a near-eyedisplay, as an example application for digitally encoded hologramscomputed using the methodology described herein. The holograms computingusing the methodology described herein can be used in any type ofdisplay, including heads-up displays for aircraft or automobiles.

FIG. 5 is a flow chart illustrating an exemplary process for digitalholography using the subject matter described herein. Digital holographyis intended to include Hogel-free holography as defined in this sectionor Wirtinger holography, as described in the sections below. Referringto FIG. 5, in step 200, a hologram is modeled using a forwardpropagation model that models propagation of a light field from ahologram plane to an image plane . For example, the propagation modelmay be a convolution, angular spectrum, Fourier, Fraunhofer, Fresnel, orany other suitable forward propagation model.

In step 202, the hologram is computed as a solution to an optimizationproblem that is based on the model. In one example, the optimizationproblem uses the least squares optimization function described abovethat minimizes an error between the square of the convolution of thehologram and the lens phase function and the square of target imageintensities. In step 204, the hologram is used to configure at least onespatial light modulator. Configuring the spatial light modulator mayinclude adjusting the pixels of a display, such as an LCD, to implementthe phases specified in the hologram function. In step 206, the spatiallight modulator is illuminated with a light source to generate thetarget image. Thus, the light source and the spatial light modulator maytogether be a holographic projector included in the head-mountabledisplay.

Wirtinger Holography for Near-Eye Displays

Near-eye displays using holographic projection are emerging as anexciting display approach for virtual and augmented reality athigh-resolution without complex optical setups-shifting opticalcomplexity to computation. While precise phase modulation hardware isbecoming available, phase retrieval algorithms are still in theirinfancy, and holographic display approaches resort to heuristic encodingmethods or iterative methods relying on various relaxations.

In this work, we depart from such existing approximations and solve thephase retrieval problem for a hologram of a scene at a single depth at agiven time by revisiting complex Wirtinger derivatives, also extendingour frame-work to render 3D volumetric scenes. Using Wirtingerderivatives allows us to pose the phase retrieval problem as a quadraticproblem which can be minimized with first-order optimization methods.The proposed Wirtinger Holography is flexible and facilitates the use ofdifferent loss functions, including learned perceptual lossesparametrized by deep neural networks, as well as stochastic optimizationmethods. We validate this framework by demonstrating holographicreconstructions with an order of magnitude lower error, both insimulation and on an experimental hardware prototype. FIG. 6 is an imageof the experimental prototype along with a comparison of imagesgenerated using Wirtinger holography, modified Gerchberg-Saxtonholography, and double-phase encoding holography.

1 Introduction

Over the last few years, near-eye display approaches have been emergingat a rapid pace, ultimately promising practical and comfortable virtualreality (VR) and augmented reality (AR) in the future. Head-mounteddisplays for stationary virtual reality use-cases have become ubiquitousconsumer products, having solved resolution, field of view (FOV),tracking and latency limitations of the past decades of research.Today's AR devices partially inherit these advances but are stilllimited to large form factors and are unable to allow for occlusions andcontinuous focus cues needed to avoid the vergence-accommodationconflict. Approaches using conventional optics to address theseindividual problems are often restricted to large setups, possiblyinvolving actuation [Chakravarthula et al. 2018]. Near-eye displaysusing light field modulation [Huang et al. 2015a; Lanman and Luebke2013] offer an alternative approach with impressive results but onlyprovide sparse ray-space sampling at acceptable spatial resolutions—thespatio-angular resolution trade-off is fundamentally limited bydiffraction.

In theory, digital holography offers an elegant way to solve the complexoptical design challenges for near-eye displays. Instead of physicallycontrolling the emitted wavefront of light with cascades of (variable)refractive optics and conventional displays, holographic displays shiftthis process to computation. Given a scene intensity image andassociated depth map, holographic display methods solve for the statesof a phase (and amplitude) modulator to encode the corresponding phasein an incident source wavefront. Assuming for a moment that the encodingprocess is perfect, this approach allows for high-resolution andcontinuous focus cues. Building on precise phase modulation hardware anda large body of existing work in holography, the recent seminal works ofMaimone et al. [2017] and Shi et al. [2017] demonstrate impressivehigh-FOV holographic near-eye displays in a light-weight wearableform-factor. Although these approaches offer insight into the promise ofholographic near-eye displays, they rely on heuristic holographic phaseencoding schemes that severely restrict the achievable image quality andfuture re-search. Akin to early image processing methods, it ischallenging to solve for different loss functions, or researchadditional modulation schemes or hybrid systems.

In this work, we deviate from heuristic solutions to holographic phaseretrieval. Instead, we rely on formal optimization enabled by complexWirtinger derivatives. With a differentiable forward model and its(Wirtinger) gradient in hand, we formulate a quadratic loss functionthat is solved via a first-order optimization method. Specifically, wedemonstrate that the problem is not required to be relaxed, as in recentlifting methods [Candes et al. 2015] but can be directly solved withstandard methods. We show that the phase retrieval problem for digitalFresnel holography can be solved efficiently using a quasi-Newton methodor stochastic gradient descent. We achieve an order of magnitude lowerreconstruction error, i.e., 10 dB peak signal to noise ratio (PSNR)improvement. We validate the proposed method in simulation and using anexperimental prototype, demonstrating that this improvement eliminatessevere artefacts present in existing methods. We validate theflexibility of the proposed phase retrieval method by modifying theobjective with a learned perceptual loss, which can be optimized usingvanilla stochastic gradient descent (SGD) methods. Moreover, we show, insimulation, that the proposed framework allows to incorporate additionaloptics or modulation in a plug-and-play fashion.

Specifically, we make the following contributions in this work:

-   -   We introduce complex Wirtinger derivatives for holographic        display formation, that allow us to directly solve holographic        phase retrieval problems as formal optimization problems using        first-order optimization algorithms.    -   We validate the proposed differentiable framework by solving        phase retrieval of holographic projections with constant focus        over the image as a least-squares problem using off-the-shelf        quasi-Newton optimization. The resulting holographic        reconstructions have an order of magnitude lower error than        previous methods.    -   We demonstrate the flexibility of the Wirtinger framework by        solving for a perceptual loss function parameterized by a        convolutional deep neural network. Moreover, in simulation, we        show that the method facilitates alternative optical setups such        as cascaded modulation on two sequential phase SLMs.    -   We assess the proposed framework experimentally with a prototype        near-eye holographic display setup. The proposed method reduces        severe artifacts of existing holographic imaging approaches.

Limitations. Although the proposed approach provides unprecedentedholographic phase retrieval quality, while being flexible as a formaloptimization framework, we do not achieve real-time frame rates onconsumer hardware. The runtime is comparable to that of the modifiedGerchberg-Saxton method from [Peng et al. 2017], at about 30 sec for afull HD hologram on a consumer laptop computer with partial GPUacceleration. However, akin to ray tracing, which recently has beenenabled at real-time rates using dedicated hardware, we hope that alow-level implementation on next-generation GPU hardware, or dedicatedhardware, could overcome this limitation in the future. In the meantime,it is practical to use a computationally cheap initial iterate combinedwith Wirtinger refinement. We anticipate the extension of Wirtingerholography to 3D holograms, defined as holograms which simultaneouslyaddress points at different depths. In addition, it is possible toextend the technique described herein to render high-quality 3Dvolumetric scenes by dynamically adjusting the focal distance to thefoveal region of interest, as demonstrated earlier by [Maimone et al.2017].

2 Related Work

Computational holograms are traditionally classified into Fourierholograms corresponding to far-field Fraunhofer diffraction, and Fresnelholograms which produce images in the near field, as determined by theFresnel Number. The computational complexity between near-field andfar-field holograms is substantial. While computing far-field hologramsusing 2D Fourier transforms can be accomplished using fast algorithmsand optimized hardware, computing near-field Fresnel holograms admits nosimple analytic solution. In fact, it is mathematically equivalent toinverting a generalized scalar diffraction integral [Underkoffler 1991].A majority of work on this topic investigates numerical techniques forgenerating Fresnel fringe patterns. In this section, we brieflysummarize prior work related to hologram computation and display.

2.1 Traditional Phase Retrieval Algorithms

A Fourier hologram produces a flat image at a far distance Fourierplane, and such a hologram is often computed using traditional phaseretrieval techniques. Phase retrieval is the method of recovering anunknown signal from the measured magnitude of its Fourier transform.Since the phase is lost in the measurement of the signal, the inverseproblem of recovering it is generally ill-posed. However, the phase canbe perfectly recovered, in theory, by solving a set of non-linearequations if the measurements are sufficiently over-sampled [Bates1982]. Early methods of phase retrieval included error reduction methodsusing iterative optimization [Gerchberg 1972; Lesem et al. 1969],together with an assumption on a non-zero support of the real-valuedsignal, with applications in optics, crystallography, biology andphysics. Extension of such iterative algorithm is the popular hybridinput-output (HIO) method [Fienup 1982], and others with variousrelaxations [Bauschke et al. 2003; Luke 2004]. Phase-retrieval methodsusing first-order non-linear optimization have been explored in the pastto characterize complex optical systems [Fienup 1993; Gonsalves 1976;Lane 1991], eventually also sparking recent work on using alternativedirection methods for phase retrieval [Marchesini et al. 2016; Wen etal. 2012], non-convex optimization [Zhang et al. 2017], and methodsovercoming the non-convex nature of the phase retrieval problem bylifting, i.e. relaxation, to a semidefinite [Candes et al. 2013] orlinear program [Bahmani and Romberg 2017; Goldstein and Studer 2018].

2.2 Computational Fresnel holograms

Fresnel hologram computation can be categorized into two classes. 1)Geometry-based techniques, which model three-dimensional scene geometryas a collection of emitters; either point-emitters (point-source method)or polygonal tiles (polygon-based method). The collective interferenceof these emitters with a reference wave is computed at a set ofdiscretized locations throughout the combined field to generate ahologram of the scene [Benton and Bove Jr 2008]. 2) Image-basedtechniques, which leverage the advantage of computer graphics renderingtechniques along with wave propagation. Next, we review geometry andimage-based methods.

Point-source methods. Waters et al. [1966] were the first to proposeusing a collection of points rather than finite sized objects to model ascene for holographic image generation. By using a look-up table ofprecomputed elemental fringes, Lucente et al. [1993] sped up thepoint-source hologram computation to under one second. Recentpoint-source based computer generated holography (CGH) computationmethods leverage the parallelization of a computer graphics card [Chenand Wilkinson 2009; Masuda et al. 2006; Petz and Magnor 2003]. Mostrecent work of Maimone et al. [2017] presents a point-source based CGHcomputation for holographic near-eye displays for both virtual andaugmented reality. All of these methods have in common that the opticaltransmission process of different primitives is modeled to beindependent, and hence, it is challenging to accurately representview-dependent mutual-occlusion and shading. Moreover, a continuousparallax demands a very dense set of point-sources which requires alarge compute budget.

Polygonal methods. Polygonal methods for computing CGH also have existedfor decades [Leseberg and Frère 1988]. The basic idea of this approachto CGH is to represent a 3D object as a collection of tilted and shiftedplanes. The diffraction patterns from a tilted input plane can becomputed by the fast Fourier transform (FFT) algorithm and an additionalcoordinate transformation. This analysis can also be done in the spatialfrequency domain by using the translation and rotation transformationsof the angular spectra permitting the use of FFT algorithms [Tommasi andBianco 1993]. Moreover, a property function can be defined for eachplanar input surface to provide texture [Matsushima 2005] and shading[Ahrenberg et al. 2008; Matsushima 2005]. Researchers also have exploredgeometric facet selection methods by ray tracing [Kim et al. 2008],silhouette methods [Matsushima and Nakahara 2009; Matsushima et al.2014] and inverse orthographic projection techniques [Jia et al. 2014]to provide occlusion culling effects.

Image-based methods. Image-based holography techniques can be broadlycategorized into light field and layer-based methods. Light fieldholograms, also known as holographic stereograms, partition a hologramspatially into elementary hologram patches, each producing local raydistributions (images) that together reconstruct multiple viewssupporting intra-ocular occlusions [Lucente and Galyean 1995; Smithwicket al. 2010; Yamaguchi et al. 1993]. Holo-graphic stereograms can bepaired with the point-source method to enhance the image fidelity toprovide accommodation and occlusion cues [Shi et al. 2017; Zhang et al.2015]. Layer-based methods, in contrast, compute the hologram by slicingobjects at multiple depths and superimposing the wavefronts from eachslice on the hologram plane [Bayraktar and Özcan 2010; Zhao et al.2015]. Layer-based and light field methods can both be combined toproduce view-dependent occlusion effects [Chen and Chu 2015; Zhang etal. 2016]. While prior work on holographic phase retrieval addressedcomputing holograms for both 2D and 3D scenes, the focus of the subjectmatter described herein lies on rendering high-quality images of scenesat a single depth plane. The method can be extended to volumetric scenesas discussed in Section 6.2.

3 Computational Display Holography

Computational display holography (CDH) simulates the physical processesof an optical hologram recording and reconstruction, using numericalmethods. The computed holographic image is brought to life by aholographic display, typically consisting of an illuminating source anda phase-modifying (sometimes also amplitude-modifying) spatial lightmodulator (SLM). The phase of the SLM describes the delay of theincident wave phase introduced by the SLM element. Note that thegeometrical (ray) optics model, commonly used in computer graphics,models light as rays of photon travel instead of waves. Although beingan approximation to physical optics, i.e. ignoring diffraction, rayoptics still can provide an intuition: the perpendiculars to the wavescan be thought of as rays, and, vice versa, phase intuitively describesthe relative delay of photons traveling along these rays. We refer thereader to [Goodman 2005] for a detailed review.

In the following, we review Fresnel holography, which can pro-ducenear-field imagery for near-eye display applications. Specifically, wereview propagation for scenes modeled as point source emitters and aconvolutional forward model, which both provide intuition for the imageformation process. The proposed method is not limited to either of thesepropagation models but, indeed, flexible to support a variety ofdifferent propagation models, including Fourier-based propagation,Fresnel, Fraunhofer, and angular spectrum propagation, all of which wereview in the supplemental material.

3.1 Point-source Propagation

Computing a hologram requires knowledge of both the illuminatingreference wave and the object wave. Given a 3D model of the scene, theobject wave can be modeled as a superposition of waves from sphericalpoint emitters at the location of object points and propagating towardsthe hologram plane where the SLM is located, see FIG. 7. The object waveE_(o) reaching a pixel at location {right arrow over (p)} on the SLM atthe hologram plane, from a point source located at an object point at{right arrow over (o)}, can be expressed as

$\begin{matrix}{{{E_{O}( \overset{arrow}{p} )} = {a_{o}\exp^{j({\phi_{o} + \frac{2\pi\;{r{({\overset{arrow}{p},\overset{arrow}{o}})}}}{\lambda}})}}},} & (1)\end{matrix}$where λ is the illumination reference wavelength, a_(o) is the amplitudeof wave at the object point {right arrow over (o)}, ϕ₀ is the (initial)phase associated with each diffuse object point and r({right arrow over(p)}, {right arrow over (o)}) is the Euclidean distance between theobject point and the SLM pixel.

Equation 1 describes the phase patterns corresponding to each objectpoint on the hologram plane. Under a paraxial approximation these phasepatterns act as Fresnel lenses, also see supplemental material. Owing tothe optical path difference, the spatial frequencies of these phasepatterns increase from the center towards the edges, where the light isdeflected maximally. However, the maximum deflection angle supported byan SLM is limited, hence restricting the lens phase patterns to asmaller region. Therefore, the overall hologram can be thought of as asuperposition of local lens phase patterns, or phase Fresnel lenses,corresponding to individual target object points

$\begin{matrix}{{{H( \overset{arrow}{p} )} = {{\sum\limits_{o \in s_{p}}{a_{o}\exp^{j({\phi_{o} + \frac{2\pi\;{r{({\overset{arrow}{p},\overset{arrow}{o}})}}}{\lambda}})}}} = {\sum\limits_{o \in s_{p}}{a_{o}{L_{o}( \overset{arrow}{p} )}}}}},} & (2)\end{matrix}$Where s_(p) is the set of object points whose phase Fresnel lenses aredefined at a pixel {right arrow over (p)}, and L is the lens phasefunction of the phase Fresnel lens for a given object point.3.2 Convolutional Propagation

The point-wise integration described in the previous paragraphs requiresintensive computation when all object points of a full three-dimensionalscene are considered. If we make the assumption that the scene islocated on a single depth plane, this integration is equivalent to theconvolution of the target image with a complex valued lens phasefunction L [Maimone et al. 2017]H=A*L,  (3)where * is the complex-valued convolution operator, A represents thecomplex amplitude of the underlying image, and L is a spatiallyinvariant lens phase function which is the same for all object points{right arrow over (o)}. Although the focus, i.e., depth, over the imageis constant, the focus may be changed globally either on a per-framebasis or in a gaze-contingent manner by employing eye tracking [Maimoneet al. 2017].

For a given hologram H from Equation 3, an approximate holographic imageon the image plane, after propagating the modulated wave field, can beformulated as a convolution of the hologram with the complex conjugateof the lens functionI=|H*L|²=|

⁻¹(

[H]∘

[L])|²,  (4)where |.|² denotes the element-wise absolute value squared operation, Lis the conjugate lens phase function,

the Fourier transform operator, ∘ is the Hadamard element-wisemultiplication and I is the intensity of the image.4 Wirtinger Holography

In this section, we present a framework for computing high-qualityphase-only holograms for near-eye display applications. We start byformulating phase hologram computation as a general optimization problemindependently of the propagation model, and only assuming that thisforward model is differentiable. Once cast as an optimization problem,we show how first-order optimization techniques can be applied to solvethis complex non-convex problem.

For the sake of brevity, the penalty functions and correspondinggradients in this section are defined for matrices in vectorized form.For the ease of notation, we use matrices and their vectorized versionsinterchangeably. Note that, in practice, we do not explicitly formmillion-dimensional vectors for optimizing holograms, but insteadperform all operations on tensors for efficient memory usage andcomputation.

4.1 Synthesizing Optimal Phase Holograms

Section 3 discussed the computation of Fresnel holograms, and thereconstruction of holographic images by propagating the field from thehologram plane to the image plane using different propagation models. Ifz is the complex wave field at the image plane, the error between thereconstructed image |z|² produced by the hologram H and the target imageI can be computed asErr=ƒ(|z| ², I)=ƒ(z),  (5)for a general penalty function ƒ. Note that the field z is a function ofthe hologram H, and is the output of the propagation function P, asdiscussed in Section 3, that isz=P(H)  (6)The phase-only hologram H has a constant amplitude across the hologramplane with each of its entries of the formH(Φ)=cexp^(jΦ)=c(h _(i))=cexp^(jϕi),  (7)where c is the (constant) amplitude, and ϕ_(i) is the phase associatedwith the i-th pixel on the SLM located at the hologram plane. Withoutloss of generality, the value of c may be set to unity for all practicalpurposes. We are interested in computing the phase-only hologram todisplay on a phase modulating SLM which can be formulated as solution tothe following non-convex error minimization problem

$\begin{matrix}{\Phi_{opt} = {{\underset{\Phi}{minimize}\mspace{14mu}\underset{{Err}{(\Phi)}}{\underset{︸}{f( {{{\mathcal{P}( {H(\Phi)} )}}^{2},I} )}}} + {\gamma{{{\nabla\Phi}}^{2}.}}}} & (8)\end{matrix}$where ∥∇ϕ∥² is an optional regularizer on the phase patterns, which may,for example, enforce constraints on the phase variations to account forSLM limitations, e.g., limited modulation range of frequency. We showthat this non-convex holographic phase retrieval problem can be solvedusing first-order optimization algorithms. In the following section, wediscuss calculating the Wirtinger gradient for the objective functionbefore describing how we apply first-order optimization methods.4.2 Computing the Complex Gradient

Chain rule. The gradient of the objective from Equation 8 can becalculated from Equations 5, 6 and 7, using chain rule as follows

$\begin{matrix}{{\frac{d({Err})}{d\;\Phi} = {\underset{I}{\underset{︸}{\frac{df}{dz}}}\mspace{14mu}\underset{II}{\underset{︸}{\frac{dz}{dH}}}\;{\underset{III}{\underset{︸}{\frac{dH}{d\;\Phi}}}.}}}\;} & (9)\end{matrix}$

Non-Holomorphic Objective. We assume that our objective is a real-valuednon-convex function of complex variables, that is ƒ:

^(n)

, where n is the number of elements in z (equal to that in H). A complexfunction that is complex differentiable at everypoint in its domain iscalled a holomorphic function. However, areal-valued function of acomplex argument cannot be holomorphic unless it is a constant—itsderivative is not defined in its domain. The derivative of such aholomorphic real-valued function is always zero, refer to supplementalmaterial. Since our objective function is not a constant, thederivatives of any order are not defined in its domain. We note that[Fienup 1993] arrived at a relatively similar analytic expression forthe gradient of an l₂-penalty, see discussion in the supplementalmaterial.

Next, we introduce Wirtinger derivatives [Remmert 2012] as a gradientapproximation that is defined. Specifically, in Section 4.3, we discusscomputing Part I of Equation 9 which is the derivative of the scalarerror with respect to the complex wave field on the image plane. Part IIof the chain rule expression in Equation 9 depends on the propagationmodel. We derive this derivative for the convolutional propagation modelin Section 4.4, to which we apply Part Ill. We derive Wirtingerderivatives for other propagation models in the supplemental material.

4.3 Wirtinger Derivatives

To overcome the undefined nature of the gradient of Part I, we use thefollowing approximation for partials of a scalar function of a complexvector ƒ:C^(n)

d(Err)=df(z)=Re

∇ƒ, dz

,  (10)where Re denotes the real part of a complex number and

., .

de-notes the inner product of two vectors. This definition, although notthe exact gradient, is consistent with the main properties of gradientsused in first-order optimization methods: 1) the gradient defines thedirection of maximal rate of change of the function, and 2) the gradientbeing zero is a necessary and sufficient condition to determine astationary point of a function. To obtain ∇ƒ, we compute the Wirtingerderivatives of the function ƒ with respect to the complex argument z.Assume ƒ(z) to be a function of two independent vectors z and z, itsWirtinger derivative can be defined as

$\begin{matrix}\begin{matrix}{{df} = {{( {\nabla_{z}f} )^{T}{dz}} + \underset{\_}{( {\nabla_{\overset{\_}{z}}f} )^{T}d\overset{\_}{z}}}} \\{= {{( {\nabla_{z}f} )^{T}{dz}} + {( {\nabla_{z}f} )^{T}{dz}}}} \\{= {2{{Re}( {( {\nabla_{z}f} )^{T}{dz}} )}}} \\{= {{Re}\langle {{2{\nabla_{\overset{\_}{z}}f}},{dz}} \rangle}}\end{matrix} & (11)\end{matrix}$We simplify the computation of ∇ƒ(z) as a scaled gradient of ƒ withrespect to only z∇ƒ(z)=2∇_(z)ƒ.  (12)Equation 12 is valid for any general scalar function of complexvariables. However, following Equation 5, we further decompose Equation12 as

$\begin{matrix}{{\nabla_{\overset{\_}{z}}f} = {{\frac{df}{\underset{A}{\underset{︸}{d( {z}^{2} )}}} \circ 2}{\underset{B}{\underset{︸}{\nabla_{\overset{\_}{z}}( {z}^{2} )}}.}}} & (13)\end{matrix}$Note that |z|² is the intensity of the reconstructed image on the imageplane. The derivative of a scalar loss with respect to the reconstructedholographic image (Part A) is defined for any differentiable penaltyfunction—even for a neural-network based loss function, where one canobtain gradients using back propagation. To plug a custom loss function,standard or neural network learned, into our framework only requires aHadamard product of the gradient of the error function with respect tothe predicted image (Part A) and the Wirtinger derivative of fieldintensity on the image plane (Part B), which is given by the scaledvalue of the field itself as2∇ _(z) (|z| ²)=2∇ _(z) (zz )=2z.  (14)

For example, for an l₂-penalty, i.e., setting ƒ(·)=∥·∥², the value of ∇ƒcan be obtained as follows

$\begin{matrix}{\begin{matrix}{{\nabla f} = {2{\nabla_{\overset{\_}{z}}f}}} \\{= {\frac{\overset{A}{\overset{︷}{d( {\frac{1}{2}{{{z}^{2} - I}}^{2}} )}}}{d( {z}^{2} )} \circ \overset{B}{\overset{︷}{2{\nabla_{z}( {z}^{2} )}}}}} \\{\mspace{20mu}{= {2{( {{z}^{2} - I} ) \circ 2}z}}}\end{matrix}.} & (15)\end{matrix}$Note: In the following sections, vec denotes the vectorization operatorfor a multi-dimensional signal, F[.] denotes a Discrete FourierTransform (DFT) operator applied to multi-dimensional signal, and Frepresents the corresponding matrix applied to the vectorized signal, inthe sense that

[H]=Fvec(H).  (16)The operator ∘ denotes the element-wise (Hadamard) product, and F^(†) isthe Hermitian (conjugate transpose) of F. In the following, we use thefact that F is unitary, and therefore F⁻¹=F^(†). We introduce the DFTmatrix F here for notational brevity. In practice, the DFT is performedby the Fast Fourier Transform (FFT) algorithm and the matrix F is neverexplicitly constructed. Note also that H is a function of Φ, the phasethat we are interested in, and FH actually means F vec(H), as shorthandnotation. All multiplications, unless specifically stated, areelement-wise multiplications.

Next, we derive the gradient for the convolutional propagation modelfrom Sec. 3.2. Please refer to the supplemental material for thegradients for other propagation models.

4.4 Gradient for Convolutional Propagation

As discussed in Section 3.2, the image on the destination plane can becomputed as I′=|H*L|². Expressing convolution as a multiplication in theFourier domain, the wave field at the destination plane can be obtainedas z=F⁻¹(F[H]∘F[L]). Deriving Part II from the partial computed inEquation 10 yields

$\begin{matrix}{\begin{matrix}{{d( {{Err}(H)} )} = {{Re}\langle {{\nabla f},{dz}} \rangle}} \\{= {{Re}\langle {{\nabla f},{d( {{F^{\dagger}({FH})}({FL})} )}} \rangle}} \\{= {{Re}\langle {{\nabla f},{F^{\dagger}{FLFdH}}} \rangle}} \\{= {{Re}\langle {{{F^{\dagger}({FL})}^{\dagger}F{\nabla f}},{dH}} \rangle}} \\{= {{Re}\langle {{{F^{\dagger}({FL})}*F{\nabla f}},{dH}} \rangle}}\end{matrix}.} & (17)\end{matrix}$

Finally, evaluating Part III with H =exp^(jΦ), we derive the partial ofthe loss function with respect to the phase Φ as follows

$\begin{matrix}{\begin{matrix}{{d( {{Err}(\Phi)} )} = {{Re}\langle {{{F^{\dagger}({FL})}*F{\nabla f}},{d( \exp^{j\;\phi} )}} \rangle}} \\{= {{Re}\langle {{{- j}\;\exp^{- {j\phi}}{F^{\dagger}({FL})}*F\;{\nabla f}},{d(\phi)}} \rangle}}\end{matrix}.} & (18)\end{matrix}$Since the phase Φ is real-valued, the above inner-product becomes thegradient, that is∇Err(Φ) =Re(−jexp^(−jΦ) F ^(†)(FL)*F∇ƒ).  (19)5 Setup and Implementation

We assess the holograms computed by the proposed method in simulationand experimentally using a hardware display prototype. We use severalloss functions, including a neural network based learned perceptualerror metric, and compute all the corresponding gradients according toSection 4.3.

5.1 Hardware Prototype

The prototype display system for experimental validation is similar tothe one demonstrated by Maimone et al. [2017] but differs in a varietyof implementation details. We use plane wave illumination in theprojection light engine. Our prototype display includes a HOLOEYE LETO-Iliquid crystal on silicon (LCoS) reflective phase-only spatial lightmodulator with a resolution of 1920×1080. The pixel pitch of the SLM is6.4 μm, resulting in the active area of 12.28 mm x 6.91 mm. We use asingle optical fiber that is coupled to three laser diodes emitting atwavelengths 446 nm, 517 nm, and 636 nm, in combination with collimatingoptics, to illuminate the SLM. The laser power is controlled by a laserdiode controller, and the output is linearly polarized as required bythe SLM. The SLM has a refresh rate of 60 Hz and is illuminated by thelaser source in a color field sequential manner.

FIG. 8 shows the optical design of the proposed prototype system. Thelight from the laser diodes is collimated using an achromatic doubletbefore being modulated by the LETO phase SLM, and then focused by a 150mm achromatic lens. An iris is placed at an intermediate image plane todiscard unwanted higher diffraction orders, especially for double phaseencoding holograms. Finally, a combination of 80 mm and 150 mmachromatic lenses is used to relay the image to the SLM sensor plane. Anoptional zero-order stop can be placed immediately after the 80 mmachromat. The optical relay lens system was optimized to have minimalaberrations, allowing a fair evaluation of different methods withwell-corrected images. To assess the display's image quality, we captureimages from the prototype holographic display using a Canon Rebel T6iDSLR camera body (without camera lens attached) with an outputresolution of 6000×4000, and a pixel pitch of 3.72 μm.

Note that achieving higher image quality in our experimental setuprequires an intermediate image plane where a zero-order stop, and otherfilter stops can be placed to filter out undiffracted and higher-orderdiffracted light. Using off-the-shelf optics to create a high-qualityimage relay resulted in a fairly long optical system. Miniaturizing oursetup to an actual head-mounted display remains an exciting area offuture research.

5.2 Implementation

We compute phase-only holograms on a PC with an Intel Xeon2.4 GHz CPUand 64 GB of RAM, and an NVIDIA GeForce GTX 1080GPU. For quadratic lossfunctions we use quasi-Newton optimization and implement the proposedmethod in MATLAB using the L-BFGS [Liu and Nocedal 1989] quasi-Newtonmethod. In particular, we use the minFunc optimization package [Schmidt2005]. For loss functions requiring stochastic gradient optimization,such as neural-network based losses, we implement the proposed method inTensorFlow [Abadi et al. 2016] and use the Adam optimizer. Specifically,we use a learning rate of 0.001 and an exponential decay rate of 0.9 forthe first moment estimate, and an exponential decay rate of 0.999 forthe second moment estimate. We obtain the gradient for the loss functioncomponent (i.e., Part I of Equation 9) from backpropagation inTensorFlow, especially for losses parameterized by convolutional neuralnetworks. We then use this gradient to compute the complex Wirtingergradient (i.e. Part II and Part Ill of Equation 9) and feed it back tothe optimizer for computing optimal phase holograms. Full colorholograms are generated by performing optimization sequentially for allthree color channels.

To compare runtimes of our Wirtinger Holography algorithm against thestate of the art modified-GS method [Peng et al. 2017] and double phaseamplitude encoding of Fresnel holograms [Maimone et al. 2017], wemeasure computation time needed to generate a 1920×1080 hologram both onMATLAB running only on CPU, and on Tensorflow with native GPU support.

MATLAB. We compute phase holograms on a per-channel basis on the CPU, inMATLAB, for all three methods. The double phase encoding method, usingonly one convolution, is the fastest with only 7 sec on MATLAB whenperformed using FFTs for a large lens kernel size. The Modified GSalgorithm takes about 95 sec for 60 iterations for generating a full HDimage. Note that increasing the number of iterations also increases thequality of GS hologram. Without warmstarting and initializing withrandom phase, our MATLAB implementation takes about 70 sec to typicallysurpass a PSNR of 30 dB in about 20 iterations.

TensorFlow. To compute phase patterns for all three colors usingTensorFlow with GPU acceleration, modified GS takes about 40 sec for 60iterations, double phase encoding method takes about 10 ms while ourmethod, initialized with random phase, takes about 30 sec beforereaching a PSNR of over 30 dB in about 200 iterations using the Adamoptimizer with the parameters discussed above.

Note that the runtimes for Wirtinger Holography is comparable or lessthan that of modified GS algorithm, and achieve significant qualityimprovement in very few iterations.

6 Analysis

In this section, we analyze the proposed method in simulation, and wevalidate its flexibility for different objective functions, forwardmodels and setup configurations in simulation. First, we compare ourapproach against the state-of-the-art holographic phase retrievalmethods from [Peng et al. 2017], which is a modified Gerchberg-Saxtonmethod (GS), and the double phase holography method (DPH) proposed in[Maimone et al. 2017]. For quantitative comparisons of the reconstructedholographic images, we evaluate peak signal-to-noise ratio (PSNR) as amean-squared error metric, and structural similarity Index (SSIM) forperceptual image quality.

FIG. 9 lists PSNR and SSIM metric evaluation results for the proposedmethod, using anl2loss and convolutional forward model, compared againstmodified GS phase retrieval, and DPH, over the followingsuper-resolution image benchmark datasets: SetS [Bevilacqua et al.2012], Set14 [Zeyde et al. 2010] and Urban100 [Huang et al. 2015b].Among these datasets, SetS and Set14 consist of natural scenes andUrban100 contains challenging urban scenes images with details indifferent frequency bands. For a fair comparison, the intensity of thereconstructed images is adjusted to have the same mean value as theircorresponding target images, as proposed in [Yoshikawa et al. 2016].

We find that our method produces superior reconstruction qualityapproaching 40 dB on an average. The proposed method outperforms doublephase encoding holograms by more than 10 dB across all datasets and themodified GS method from [Peng et al. 2017] by more than 20 dB onaverage. Also note that the SSIM measurements validate that our methodproduces reconstructions with perceivably fewer errors as compared todouble-phase holograms. FIG. 10 shows qualitative comparisons forselected image patches. Additional examples are provided below in thesection entitled Supplementary Information for Wirtinger Holography forNear-eye Displays.

Resolution. The double phase encoding method from Maimone et al. [2017]produces a good approximation of the target image. However, encoding asingle complex value into two phase values results in loss of resolutionin holographic images. In contrast, our method produces high qualityreconstructions without significant loss of resolution. For example,notice the enhancement of text in FIG. 10 (rows 2 and 3), and thereproduction of details (3rd row).

Ringing. Ringing is a ripple artefact that occurs near sharp edges.Although ringing artefacts from double phase encoding are subtle forsmooth images, they are prominent in images with high frequency details.These artefacts are particularly apparent as they can differ in thecolor channels, resulting in chromatic ringing. The last row of FIG. 10shows an extreme case where ringing significantly degrades theholographic image quality produced by a double phase hologram. Incontrast, the proposed method produces a faithful reconstruction of theimage.

Noise and Contrast. Compared to the modified GS method from Peng et al.[Peng et al. 2017], the proposed method suffers from significantly lessreconstruction noise, an artefact type typical to Gerchberg-Saxton-stylemethods. Moreover, Wirtinger Holography allows for improved contrast andreduced chromatic artefacts due to the improved quality across all colorchannels.

6.0.1 Alternative Penalty Functions. To validate the flexibility of theproposed framework, we show results for three different penaltyfunctions for the proposed setup and the convolutional forward model. Weevaluate the following penalty functions (see Sec. 4.1):1) l₂-normleast-squares objective 2) MS-SSIM [Wang et al. 2003] error minimizationas a hand-crafted perceptual accuracy 3) the learned perceptualsimilarity metric [Zhang et al. 2018] as a deep neural networkperceptual quality metric.

FIG. 11 shows evaluations on a custom dataset of 20 images randomlypicked from Set5, Set14, Urban100 and BSDS100 [Martin et al. 2001]datasets to cover images with a range of spatial frequencies. Optimizingfor hand-crafted or learned perceptual loss metrics improves therespective metric value by a significant margin over a least-squaresloss function. The margin between the learned LPIPS loss and theleast-squares loss is the largest with about 3.5 dB or 0.02 in SSIM.Note that the LPIPS loss function offers the best performance in allthree image quality metrics, illustrating the promise of our flexibleframework for future research on formal optimization for phaseretrieval. The qualitative examples on the top in FIG. 11 showperceptually strong differences especially in uniform and smoothregions, where localized errors average out in the least-squaresestimate but are penalized by the learned perceptual metric.

6.1 Generalization to Alternative Optical Designs

To demonstrate the ability of the proposed method to generalize to otheroptical setups, we also simulate a stacked setup of two cascaded SLMs,illustrated in FIG. 12. For conventional transmissive displays, stackingtwo displays with a lateral offset of half a pixel has allowedresearchers to synthesize images with greater spatial resolution thanthat afforded by any single SLM [Heide et al. 2014]. We apply this ideato holographic display generation. Instead of calculating a spatiallysuper-resolved hologram and decomposing it into two stacked layers usingcomplex matrix factorization [Peng et al. 2017], our frameworkfacilitates end-to-end optimization of cascaded holograms.

A cascade of two phase-only modulators combine additively in phase, ormultiplicatively in their complex wave fields. Optically cascading afirst SLM (A) with phase ϕ_(A) of M pixels, and a second SLM (B) withphase ϕ_(B) of M pixels, results in a complex wave field that can beexpressed as:H=exp^(jΓ) ^(AB) ^(Φ) ^(A) ∘exp^(jΓ) ^(BA) ^(Φ) ^(B) =exp ^(j(Γ) ^(AB)^(Φ) ^(A) ^(+Γ) ^(BA) ^(Φ) ^(B) ⁾,  (20)where Γ_(AB) and Γ_(BA) are sparse transformation matrices of size 4 M×Mthat model the relative pixel shifts of the two phase-SLMs with respectto a 2× higher-resolution virtual SLM with 4 M pixels. Specifically,Γ_(AB) maps one pixel from SLM A to four pixels from the virtual SLM,and similarly Γ_(BA) represents the mapping for SLM B. For a half-pixelshift between both SLMs, these two matrices are binary matrices. Theabove equation can be easily plugged into our forward model, and theindividual SLM phase patterns can be computed as

$\begin{matrix}{\Phi_{{opt},A},{\Phi_{{opt},B} = {\underset{\Phi_{A},\Phi_{B}}{minimize}\mspace{14mu}{f( {{{\mathcal{P}( \exp^{j{({{\Gamma_{AB}\Phi_{A}} + {\Gamma_{BA}\Phi_{B}}})}} )}}^{2},I} )}}}} & (21)\end{matrix}$We simulate two reflective mode phase-only SLMs with the specificationsof the HOLOEYE LETO used in our hardware prototype, but with half theresolution (i.e. 960×540) in both dimensions to generate a hologram offour times the size of any SLM (i.e. 1920×1080). FIG. 12 shows simulatedreconstruction of images produced by a dual-SLM cascaded holographicdisplay. Notice that the cascaded hologram spatially superresolves theimage by a factor of four, i.e. twice as many pixels along each axis.The cascaded hologram offers fine detail with high contrast. While thisapproach effectively promises next generation results with currentlyavailable SLM hardware, we note that, in practice, alignment andcalibration of the overlap might be challenging without micron-accuratemechanical alignment methods, and hence require significant engineeringeffort to enable efficient manufacturing processes.6.2 3D Holograms

Although the focus of the proposed method is on improving the quality of2D holographic images, our optimization framework can be extended to 3Dvolumetric scenes. One can slice the 3D scene into multiple depth planesand superpose all complex holograms corresponding to each depth plane,thereby forming a true 3D hologram. However, this method is not optimaland causes slow down of a factor proportional to the number of depthplanes, and also will not support smooth depth changes. One option is togenerate 2D holograms approximating spatially variant focus by fastdepth switching. With gaze tracking, we can measure the depth at whichthe user is looking and compute the complex 2D hologram by changing thefocus of the scene to that depth, providing precise focus in the fovearegion [Maimone et al. 2017]. In other words, we can render a densefocal-stack in lieu of computing the full 3D holo-gram, therebyaddressing the vergence-accommodation conflict (see FIG. 16). Since theeye can focus only at a single depth any given time, presenting depth offield blur for points close to each other within the fovea region can beaddressed in the image space [Cholewiak et al. 2017]. Please refer tosub-section 3 of the Supplementary Information section below.

7 Assessment

FIG. 13 and FIG. 14 show experimentally acquired results from theprototype system presented in Section 5. We validate the proposed methodby comparing different holographic phase retrieval methods in full colorand for individual single-wavelength channels.

The results in FIG. 13 demonstrate that the proposed approach suppressesmost of the severe artefacts present in existing methods. Specifically,reconstruction noise, as in the flat areas for the GS holograms, andringing, as for the periphery in the TV line chart, limit the achievedresolution of previous methods. GS phase retrieval preserves high lightefficiency, while its iterative projection of a random phase update onthe image plane end inevitably causes severe reconstruction noise. TheDPH method leads to the unwanted effect of a noticeable portion of lightescaping the designated area. Moreover, it creates multiple higher-ordertwin images and suffers from severe ringing around high-contrast edgesand in the periphery, which are particularly visible in the binary imagein the second row of FIG. 13. The proposed Wirtinger holography methodrecovers fine detail while suppressing severe ringing and reconstructionnoise, resulting in improved resolution validated in the TV line chartareas in the top row of FIG. 13.

The full color results in FIG. 14 confirm the trend of the singlechannel reconstructions. The proposed approach reduces ringing andreconstruction noise present in existing methods. As a result, finedetails, as for the tiles on the roof of the houses or the microtubulesin FIG. 14, are displayed more faithfully with reduced chromaticartefacts. While this validates the proposed method, the margin comparedto previous methods is not on par with the simulation results. This isdue to residual artefacts resulting from a number of limitations of ourspecific low-cost prototype, which we note are not fundamental to theproposed method. As our prototype SLM is unsupported due to its age, wewere only able to rely on the available look-up-tables, which did notmatch the wavelengths of our laser configuration. This results ininaccurate phase delays which severely affects methods relying onaccurate forward models, such as the GS and proposed method. Moreover,the absence of a zero-order stop affects all methods uniformly in theproposed setup. Refer to the supplement for additional analysis. We notethat even with these prototype limitations the proposed methodoutperforms existing holographic phase retrieval methods.

8 Discussion and Conclusion

We have presented a phase retrieval method for holographic near-eyedisplays, which departs from existing heuristic algorithms, and insteadrelies on formal optimization using complex Wirtinger derivatives. Thisframework has allowed us to formulate holographic phase retrieval as aquadratic optimization problem that can be solved using first-orderoptimization methods. We have validated that the resulting hologramshave an order of magnitude lower error, i.e., more than 10 dB PSNRimprovement, when compared to recent state-of-the-art approaches. Wehave assessed the proposed method in simulation and using anexperimental prototype system. Wirtinger holography eliminates thesevere artefacts present in existing methods, and, in particular,enables us to minimize ringing and chromatic artefacts.

Although the proposed method provides unprecedented accuracy, itsruntime is only comparable to that of the modified Gerchberg-Saxtonmethod [Peng et al. 2017], with about 30 sec for a full HD hologram on aconsumer laptop computer. However, note that we implemented ourframework using high-level languages, without manual low-level codeoptimization, to allow for rapid adoption by researchers. Moreover, wedo not use dedicated hardware for our holographic display algorithm. Inthe future, we envision efficient implementations or dedicated displayhardware to overcome these limitations and achieve real-time framerates, similar to how recent dedicated hardware made real-time raytracing possible.

In addition to reducing error by an order of magnitude when compared toexisting methods, we have validated the flexibility of the proposedsystem. Specifically, we have shown that Wirtinger holographyfacilitates the use of almost arbitrary penalty functions in aplug-and-play fashion, even including learned perceptual lossesparametrized by deep neural networks, the use of different first-orderoptimization methods, including stochastic optimization methods, andthat the proposed framework can be applied to different optical setupconfigurations. As such, Wirtinger holography paves the road towardshigh-quality artefact-free near-eye holographic displays of the futureand enables future research towards this vision.

Supplementary Information on Wirtinger Holography for Near-Eye Displays

1 Introduction to Wirtinger Calculus

In this section, we provide a quick introduction to Wirtinger calculusand Wirtinger derivatives in the context of optimization problems whichinvolve derivatives of a complex variable. However, we refer interestedreaders to [Remmert 2012] for a detailed discussion on complex functionsand analysis. Consider a real-valued function of a real variable, forinstance:ƒ:

x

y=ƒ(x)∈

,  (1)the point x_(opt), where ƒ(x) is maximum (or minimum) is obtained bytaking the derivative of ƒ concerning x and setting it zero. In otherwords, for x_(opt), the following equation has to be valid

$\begin{matrix}{ \frac{df}{dx} |_{x_{opt}} = 0.} & (2)\end{matrix}$Note that here we assume ƒ(x) to be continuous in some domain

, and the derivative to exist. However, whether the obtained solutiongives a maximum or minimum value needs to be checked using additionalconditions or higher-order derivatives.

We can define the derivative for a complex function of a complexvariable, for instanceƒ:

z

w=ƒ(z)∈

,  (3)as follows

$\begin{matrix}{{f^{\prime}( z_{0} )} = { \frac{df}{dz} |_{z_{0}} = {\lim\limits_{zarrow z_{0}}{\frac{{f(z)} - {f( z_{0} )}}{z - z_{0}}.}}}} & (4)\end{matrix}$

If ƒ′(z) exists in the domain

⊂

, the function ƒ(z) is called ananalytic or a holomorphic function in

.

Now, a given complex function can be decomposed into two real functions,each depending on two real variables, say x and y, which are the realand imaginary parts of the complex variable z. Mathematically, this canbe represented asƒ(z)=ƒ(x+jy)≡u(x, y)+jv(x, y);z=x+jy  (5)

It can be shown that for the above function ƒ(z) to be holomorphic, thecorresponding component functions u(x,y) and v(x,y) need to satisfy theCauchy-Riemann conditions [Remmert 2012]

$\begin{matrix}{\frac{\partial{u( {x,y} )}}{\partial x} = \frac{\partial{\upsilon( {x,y} )}}{\partial y}} & (6) \\{\frac{\partial{\upsilon( {x,y} )}}{\partial x} = {- \frac{\partial{u( {x,y} )}}{\partial y}}} & (7)\end{matrix}$

If the above conditions are valid, then the complex derivative of aholomorphic function ƒ(z) can be expressed by the partial derivatives ofthe component real functions u(x, y) and v(x, y) as

$\begin{matrix}{\frac{{df}(z)}{dz} = {\frac{\partial{u( {x,y} )}}{\partial x} + {j\frac{\partial{\upsilon( {x,y} )}}{\partial x}}}} & (8)\end{matrix}$1.1 Real Functions of Complex Variables

As mentioned in the main manuscript, it can be seen from Equation 8 thatif the function ƒ(z) is a real valued function, then the componentfunction v(x, y) which forms the imaginary part of ƒ(z) is zero, i.e.v(x, y)=0. Following Cauchy-Riemann conditions as described in Equation6, we see that

$\begin{matrix}{\frac{\partial{u( {x,y} )}}{\partial x} = {\frac{\partial{\upsilon( {x,y} )}}{\partial y} = 0}} & (9)\end{matrix}$Thus, u(x, y) is always a constant, and so is the function ƒ(z), and thederivative of such a function is always zero. In other words, a realfunction of complex variable is not holomorphic unless it is a realconstant.

To work around this, instead of treating ƒ(z) as a real function of acomplex variable, we regard ƒ(z)=u(x, y) as a function of two realvariables. In the context of solving an optimization problem, it can betreated as a multidimensional real function. Therefore, to find theoptimum value of ƒ(z), we want to find the optimum of u(x, y), whichrequires

$\begin{matrix}{\frac{\partial{u( {x,y} )}}{\partial x} = {{0\mspace{14mu}{and}\mspace{14mu}\frac{\partial{u( {x,y} )}}{\partial y}} = 0}} & (10)\end{matrix}$Note that v(x, y)=0 since the function is real valued.

Both of the above real-valued equations for the optimal componentsx_(opt) and y_(opt) can be linearly combined into a more compact onecomplex-valued representation as follows

$\begin{matrix}{{{\alpha_{1}\frac{\partial{u( {x,y} )}}{\partial x}} + {{j \cdot \alpha_{2}}\frac{\partial{u( {x,y} )}}{\partial y}}} = {{0 + {j\; 0}} = 0}} & (11)\end{matrix}$where α1 and α2 are arbitrary real valued non-zero constants. Note thatthe above representation is valid since the real and imaginary parts areorthogonal, and the linear combination is only intended to devise acompact representation.

Let us now define a differential operator writing the real and imaginaryparts of z=x+jy as the tuple (x,y)

$\begin{matrix}{\frac{\partial\;}{\partial z} = {{\alpha_{1}\frac{\partial}{\partial x}} + {{j \cdot \alpha_{2}}{\frac{\partial}{\partial y}.}}}} & (12)\end{matrix}$Note that the operator defined in Equation 12 can also be applied tocomplex functions, because real cost functions can often be decomposedinto complex components, for example ƒ(z)=|z|²=z.z=ƒ₁(z)ƒ₂(z), whereƒ₁(z)ƒ₂(z)∈

. Also, observe that z=x−jy the complex conjugate of z=x+jy.

The choice of values α1=½ and α2=−½ meets the requirements for definingthe complex derivatives, and it was first introduced by WilhelmWirtinger. Wirtinger derivatives of a (complex) function ƒ(z) of acomplex variable z=x+jy are defined as the following linear partialdifferential operators of the first order

$\begin{matrix}{\frac{\partial}{\partial z} = {{\alpha_{1}\frac{\partial}{\partial x}} + {{j \cdot \alpha_{2}}{\frac{\partial}{\partial y}.{and}}}}} & (13) \\{\frac{\partial f}{\partial\overset{\_}{z}} = {\frac{1}{2} \cdot ( {\frac{\partial f}{\partial x} + {j \cdot \frac{\partial f}{\partial y}}} )}} & (14)\end{matrix}$For functions of many variables, ƒ:

^(n)

z=[z₁, z₂, . . . , z_(n)]^(T)

w=ƒ(z) ∈

, all n partial derivatives with respect to the complex variables z₁, .. . , z_(n) need to be calculated. In other words, the derivatives needto be calculated and combined into a vector again, and this gradient ofthe function can be defined as follows

$\begin{matrix}{\frac{\partial f}{\partial z} = {{\begin{bmatrix}\frac{\partial f}{\partial z_{1}} \\\frac{\partial f}{\partial z_{2}} \\\vdots \\\frac{\partial f}{\partial z_{n}}\end{bmatrix}\mspace{14mu}{and}\mspace{14mu}\frac{\partial f}{\partial\overset{\_}{z}}} = \begin{bmatrix}\frac{\partial f}{\partial{\overset{\_}{z}}_{1}} \\\frac{\partial f}{\partial{\overset{\_}{z}}_{2}} \\\vdots \\\frac{\partial f}{\partial{\overset{\_}{z}}_{n}}\end{bmatrix}}} & (15)\end{matrix}$Of course, the Wirtinger gradients defined in Equation 15 has to beequal to the zero vector 0=[0,0, . . . ,0]^(T). Note that for suchmultidimensional functions, one can separate and inspect individuallythe real and imaginary parts. But using the above definition, we can usesimple arithmetic rules that can be expressed using vectors andmatrices. This flexibility allows us to use formal optimization methodseasily.2 Additional Discussion on Image Formation Models

In this section, we describe additional propagation models which can beincorporated by the proposed method.

2.1 Fourier Transform-based Propagation

A further simplification, in addition to the convolutional approximationfrom the main manuscript, is a parabolic approximation that can be madeto remove the square root operation in the Euclidean distance term inEquation 2 in the main manuscript to obtain a more convenient Fresnelscalar diffraction model. However, this propagation constrains thesampling on the image plane, which is determined by the wavelength andpropagation distance from the hologram. Also, the sampling pattern onthe image plane is restricted by the sampling pattern on the hologramplane, and vice-versa. This constrained sampling is an issue when thehologram and image planes are of different sizes, especially fornear-eye displays which may use spherical wave illumination forsupporting wide field-of-view [Maimone et al. 2017; Shi et al. 2017].

One can overcome the sampling limitation by employing a two-step Fresnelpropagation using an intermediate observation plane to effectivelydecouple the sampling on both hologram and image planes [Okada et al.2013]. For a sufficiently large distance of the intermediate plane fromthe hologram plane, we can further simplify the computation by employingFraunhofer diffraction. The Fraunhofer-Fresnel two-step propagationresults in optical wave field on the image plane as obtained by:

$\begin{matrix}{{{U_{I}( {x,y} )} = \underset{E_{3}}{\underset{︸}{\frac{1}{\lambda^{2}{D( {L + D} )}}\exp\{ {\frac{jk}{2( {L + D} )}\lbrack {x^{2} + y^{2}} \rbrack} \}}}}{{\mathcal{F}\{ {\underset{E_{2}}{\underset{︸}{\exp\{ {\frac{jk}{{- 2}D}\lbrack {\zeta^{2} + \eta^{2}} \rbrack} \}}}\mathcal{F}^{- 1}\{ {U_{H}( {\zeta,\eta} )} \}\underset{E_{1}}{\underset{︸}{\exp\{ {\frac{jk}{2( {L + D} )}\lbrack {\zeta^{2} + \eta^{2}} \rbrack} \}}}} \}},}} & (16)\end{matrix}$where L and D are distances to the image plane and intermediate planefrom the hologram plane respectively, k is the wave number, U_(H)(ζ, η)and U_(I)(x,y) are the optical wave fields on the hologram and imageplane respectively. For brevity, the above equation can be expressed as:U_(I)=(x, y)=E ₃∘

(E ₂∘(

⁻¹(U_(H)))∘E ₁).  (17)2.2 Gradient for Fourier-transform Based Propagation

We discussed in the previous paragraphs that the field in thedestination plane can be obtained as z=E₃∘

(E₂∘(

⁻¹U_(H))∘E₁) for a Fraunhofer-Fresnel two-step propagation. Followingsteps as detailed in the main manuscript for the gradient of theconvolutional model, the gradient of the loss function Err(Φ) concerningΦ can be computed as:∇Err(Φ)=Re(−je ^(−jϕ) FE ₁ *E ₂ *F ^(†) E ₃*∇ƒ).  (18)2.3 Band-limited Angular Spectrum Propagation

The diffractive field from the SLM, if Fourier-analyzed across anyplane, can be identified as plane waves traveling in differentdirections away from the hologram plane. The field amplitude across anypoint can be calculated as the summation of contributions of these planewaves, taking into account the phase shifts undergone during thepropagation [Goodman 2005]. This is called the angular spectrumpropagation of the wave field. The angular spectrum method

(ASM) is equivalent to the Rayleigh-Sommerfeld solution and yieldidentical predictions of the diffracted wave field [Shen and Wang 2006].The ASM propagated field U₂(x,y;z) from a diffracting aperture with aninput source field U₁(x,y;0)can be computed as:U_(s)(x,y;z)=

⁻¹(

U₁(x,y;0)∘H).  (19)where H is the ASM transfer function given by:

$\begin{matrix}{{H( {f_{x},{f_{y};z}} )} = \{ {\begin{matrix}{{\exp\lbrack {j\; 2\pi\frac{z}{\lambda}\sqrt{1 - ( {\lambda\; f_{x}} )^{2} - ( {\lambda\; f_{y}} )^{2}}} \rbrack},} & {\sqrt{f_{x}^{2} + f_{y}^{2}} < \frac{1}{\lambda}} \\{0,} & {otherwise}\end{matrix}.} } & (20)\end{matrix}$Further, the angular spectrum transfer function can be band-limited toavoid numerical errors due to the sampling [Matsushima and Shimobaba2009].

As discussed in previous paragraphs, the wave field at the destinationplane can be obtained as z=F⁻¹(FU₁∘H), and the image is computed as|z|². Computing Part 2 from the Part 1 gradient computed in Equation 9in the main manuscript yields:

$\begin{matrix}{\begin{matrix}{{d( {{Err}(H)} )} = {{Re}\langle {{\nabla f},{dz}} \rangle}} \\{= {{Re}\langle {{\nabla f},{d( {{F^{\dagger}(H)}( {FU}_{1} )} )}} \rangle}} \\{= {{Re}\langle {{\nabla f},{F^{\dagger}{HFdU}_{1}}} \rangle}} \\{= {{Re}\langle {{F^{\dagger}H^{\dagger}F{\nabla f}},{dH}} \rangle}} \\{= {{Re}\langle {{F^{\dagger}H*F\;{\nabla f}},{dH}} \rangle}}\end{matrix}.} & (21)\end{matrix}$

Finally evaluating Part 3 with U₁=exp^(jΦ), we derive the definition ofgradient of the loss function with respect to the phase Φ as follows:

$\begin{matrix}{\begin{matrix}{{d( {{Err}(\Phi)} )} = {{Re}\langle {{F^{\dagger}H^{*}F{\nabla f}},{d( e^{j\;\phi} )}} \rangle}} \\{= {{Re}\langle {{{- {je}^{{- j}\;\phi}}F^{\dagger}H^{*}F{\nabla f}},{d(\phi)}} \rangle}}\end{matrix}.} & (22)\end{matrix}$Since the phase Φ is real valued, the above inner-product definition canbe read as:d(Err(Φ))=

Re(−je ^(−jΦ) F ^(†) H*F∇ƒ),d(ϕ)

, ∇Err(Φ)=Re(−je ^(−jϕ) F ^(†) H*F∇ƒ).  (23)

2.5 Comparison with Other Gradient Definitions

Note that the Wirtinger gradient definitions for various propagationmodels are coherent with the other definitions. For example, we considerthe gradients derived by Fienup [Fienup 1993] for an l2 error metric.For the squared difference in intensities (squared magnitudes), thederivative with respect to a parameter p is defined as follows

$\begin{matrix}{{\frac{\partial E}{\partial p} = {4{{Re}( {\sum\limits_{x_{1}}{\frac{\partial{{\mathcal{g}}( x_{1} )}}{\partial p}\{ {P^{\dagger}\lbrack {{{{G(u)}^{2}}{G^{*}(u)}} - {{{F(u)}}^{2}{G^{*}(u)}}} \rbrack} \}^{*}}} )}}},} & (24)\end{matrix}$where g(x) is the optical wavefront immediately after the input plane,G(u) is the wavefront at the detector plane, P is the propagationfunction and |F(u)| is the magnitude of the complex optical field at thedetector. The above equation can be reformulated as

$\begin{matrix}{\frac{\partial E}{\partial p} = {{{Re}( {\sum\limits_{x_{1}}{\frac{\partial{{\mathcal{g}}( x_{1} )}}{\partial p}\{ {P^{\dagger}\lbrack {2{( {{{G(u)}^{2}} - {{F(u)}}^{2}} ) \cdot 2}{G^{*}(u)}} \rbrack} \}^{*}}} )}.}} & (25)\end{matrix}$It can be observed that the above formulation is consistent with theWirtinger derivatives defined in the main manuscript, where[2(|G(u)²|−|F(u)|²)·2 G(u)] form Part 1 of Equation 9 defined in themain manuscript. Applying inverse propagation to the above equation andtaking its conjugate becomes Part 2 of the chain rule, and finallymultiplying with

$\frac{\partial{g(x)}}{\partial p}$for any parameter p; in our case phase ϕ, where g(x)=exp(jϕ); completesPart 3 of Equation 9.3 Optimizing for 3D Holograms

The proposed Wirtinger holography framework can be extended to computing3D holograms in a multi-layered or multi-focal approach. Given a 3Dmodel or the scene data, one can voxelize and remap it into multipledepth planes. The hologram corresponding to each depth plane can becomputed separately and all holograms can be later superposed togenerate a complex hologram of the 3D scene. However, such an approachmight not result in the optimal 2D phase pattern that generates theunderlying 3D scene. Another approach is to jointly optimize for alldepth planes to generate the corresponding optimal 2D hologram, which,however, is a challenging problem, as we will highlight next.

As shown in FIG. 15, let us suppose we have to create a few brightvoxels and a few dark voxels in a 3D volume. When the SLM is illuminatedby a collimated beam, the SLM pixels diffract light to focus at pointsin space to create the bright voxels. Observe that a dark voxel thatescapes any of the diffracted light paths are not difficult to create.However, it is difficult or impossible to create a dark spot in spaceimmediately after creating a focused bright spot. This requires thephase of light to change very rapidly in between the bright and darkvoxels, which is physically infeasible for a free space propagation.Therefore, simultaneously creating a 3D volume with rapidly varyingintensity is a difficult problem. However, one can generate good quality3D holograms for relatively less complicated scenes, but most often, all3D holograms suffer from noise and loss in contrast due to theabove-mentioned problem.

Computing high quality holograms by jointly optimizing for all depthplanes using our Wirtinger holography framework also suffer theabove-mentioned problem, since if the depth planes are densely spaced,the optimizer forces light to change its phase rapidly whilepropagating, to simultaneously create closely spaced bright and darkvoxels which are sometimes physically not feasible. Therefore theoptimal phase hologram that is generated might contain noise which candegrade the image quality in comparison to 2D holograms. Although all 3Dholographic rendering methods suffer from these problems, the hologramscomputed using full Fresnel diffraction integral without paraxialapproximation and heuristic encoding schemes might produce fewerartifacts than optimizing for dense focal stack of 3D scene data.However, since the eye can only focus at one focal plane at a time, onecan render holographic imagery at the eye accommodation plane bychanging the scene focus globally. This needs to be accompanied by aneye tracker. Note, that although the depth detection capabilityprogressively degrades with retinal eccentricity [Kim et al. 2017],object points which are closely spaced in the region where is eye isfoveated suffer if not presented with an accurate depth of field blur[Maimone et al. 2017]. However, such regions can be addressed byproviding the depth of field blur in the image space [Cholewiak et al.2017]. FIG. 16 shows simulated hologram results generated by ourframework for a 3D scene by the method of dynamic global scene focus.Notice that the depth of field blur is appropriately represented withoutcompromising on the image quality. Jointly optimizing for all depthplanes to generate an artifact-free full 3D hologram remains an excitingdirection for future work.

4 Additional Prototype Details

We built our bench-top prototype using cage system mounting. The opticsare adjusted for an image plane distance of about 200 mm from the SLM. ACanon Rebel T6i DSLR camera is used to capture the holograms, withoutany camera lens present, but instead using a combination of twodiffractive achromatic lenses as an eyepiece. FIG. 17 shows the layoutof our optical design.

We use red, green, and blue single-mode fiber lasers that are controlledby a Thorlabs LDC205C Laser Diode Controller, sequentially changing thethree colors. The exposure of the imaging camera and the laser intensityare adjusted once based on the laser power. All capture settings arekept constant per experiments.

5 Additional Analysis of Implementation

5.1 Initialization and Termination criteria

Here we discuss the initialization and termination criteria for theWirtinger holography framework.

To solve Equation 8 of the main manuscript, we start the optimizationprocess with an initial guess of the phase value ϕ, which could eitherbe a random phase or a carefully crafted phase. As mentioned in the mainmanuscript, carefully initializing the optimizer results in fasterconvergence compared to random (or shallow random) initialization. Itcan be easily understood from the main manuscript that the optimizercontinuously searches for the optimal phase hologram for a given targetintensity (image). However, one must note that there can be severalphase holograms that can produce a given target intensity, making ourobjective function an extremely non-convex problem with several localminima. Therefore, initializing with a random phase distribution,intuitively, expands the search space for determining an optimal phasehologram. Owing to a larger domain of possible phase holograms, theparticular choice of initial phase value may affect the convergencespeed and numerical stability.

Let us now consider one pixel on the hologram (SLM) plane. The complexamplitude of the hologram pixel constitutes a unit amplitude and a phasevalue lying in the range [0, 2 π]. In other words, the complex amplitudeof the hologram pixel can lie anywhere on a unit circle. Suppose if thephase is known to lie within a certain range [α, β], the correct complexnumber determining the complex amplitude of the hologram pixel must lieonan arc of the unit circle which is defined by the angles α and β.

In our case, the possibility of the range of phase values for eachhologram pixel remains unclear. However, given the knowledge of someinitial phase value producing the target intensity pattern, it isreasonable to assume the optimal phase value might lie somewhere closeto the initial phase value. Initializing the optimizer with a carefullycrafted phase, for example, a few iterations of Gerchberg-Saxtonalgorithm or double-phase encoding, thus offers the promise of findingoptimal phase faster. However, note that even with this additionalinformation, the problem remains non-convex and cannot directly beefficiently solved. This can be reflected in the runtime values reportedin the main manuscript, that careful initialization offers about 10-15 sof speedup, but still takes a reasonable amount of time to achieveconvergence. FIG. 18 shows the PSNR values with iteration count for ourframework for a randomly initialized phase, and it can be observed thatthe PSNR improves very quickly initially, and then slowly increases withan increasing number of iterations. For the results reported in the mainmanuscript, we terminate the optimizer once the PSNR values reachconvergence.

5.2 Additional Simulation Results

As mentioned in the main manuscript, Wirtinger holography eliminatesmuch of the ringing artifacts and generates high quality holographicreconstructions in comparison to the recently demonstrated modified GSand double phase encoding phase hologram generation algorithms. FIG. 19compares in simulation double phase encoding phase holograms andWirtinger holograms, and present SSIM error maps as an indication of theprobability of perceptually detecting ringing errors in thereconstructed image.

5.3 Cascaded Holograms

In the main manuscript, we show that a cascaded SLM setup can generatesuper resolved holographic images. FIG. 20 presents simulated imagescomparing holographic reconstructions of Wirtinger holograms obtainedfrom a cascade of two low resolution 960×540 phase-only SLMs, with thesimulated holographic images of both double-phase and Wirtingerholograms obtained from a single high resolution 1920×1080 phase-onlySLM. Note that the holographic reconstruction obtained from a cascade oftwo low-resolution SLMs is super resolved in comparison to theholographic image obtained from a high-resolution double phase hologram.However, Wirtinger holograms from a single high-resolution SLM is betterin quality compared to the holographic images from the cascadedlow-resolution SLM.

5.4 Oversampled Phase Retrieval for Imaging

To further validate the flexibility of the proposed method, we alsoapply it to phase retrieval for imaging in Table1. We adopt the4×oversampled phase retrieval problem setting discussed in [Metzler etal. 2018] without noise added in our case. We compare the baselinemethods discussed in [Qian et al. 2017] and [Metzler et al. 2018] on aset of 10 standard images. Please see both references for additionaldetails on the individual methods.

TABLE 1 Oversampled Phase Retrieval for Imaging Method PSNR [dB] HIO35.52 AltGD 35.85 TWF 35.33 MTWF 35.33 BSG 35.33 This method 36.346 Additional Experimental Validation6.1 Zero-order Diffraction Beam

We use Holoeye LETO-1 phase-only Spatial Light Modulator (SLM) with a6.4 μm pixel pitch and a 93% pixel fill factor. The fill factordescribes the size of the non-functional dead area between two adjacentpixels of the SLM. The light that is incident on these dead areas is notmodulated by the SLM and hence results in a zero-order diffraction beam(ZOD) on the optical axis. This ZOD introduces a high intensityillumination that distorts the desired light profile and undermines thehologram reconstruction quality.

A common solution to bypass the ZOD effects is to shift the illuminationaway from the optical axis, but this reduces the diffraction efficiency.To maintain high diffraction efficiency, we use an on-axis illuminationin our hardware setup. FIG. 21 presents images from our prototypedisplay highlighting the zero-order diffracted light across all threecolor channels. Note that this zero-order pattern is a constant patternacross all the holographic images, causing deterioration in the imagequality. As shown in FIG. 22, if the ZOD is eliminated, the quality ofWirtinger holograms improve significantly.

6.2 SLM Phase Patterns

We show in FIG. 23 the SLM phase patterns for the green laser of 517 nmused in our experimental setup for the phase-only hologram resultspresented in the main manuscript.

6.3 Hardware Prototype Results

We present several holographic images reconstructed on our prototypeholographic display, other than the ones presented in the mainmanuscript, in FIG. 24 and FIG. 25. Wirtinger holograms feature moredetail, higher resolution, and higher light efficiency.

The disclosure of each of the following references is herebyincorporated herein by reference in its entirety.

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It is understood that various details of the presently disclosed subjectmatter may be changed without departing from the scope of the presentlydisclosed subject matter. Furthermore, the foregoing description is forthe purpose of illustration only, and not for the purpose of limitation.

What is claimed is:
 1. A method for digital holography, the methodcomprising: modeling a hologram using a forward propagation model thatmodels propagation of a light field from a hologram plane to an imageplane; computing the hologram as a solution to an optimization problemthat is based on the model, wherein computing the hologram as a solutionto an optimization problem includes evaluating holograms as potentialsolutions to the optimization problem by computing an approximation of agradient of a penalty function with respect to amplitude of a complexwave field of a reconstructed image on the image plane using Wirtingerderivatives, wherein the penalty function quantifies a differencebetween the amplitude of the complex wave field of the reconstructedimage on the image plane and an amplitude of a complex wave field of thetarget image on the image plane, adjusting a phase of a spatial lightmodulator based on the penalty function, wherein computing anapproximation of the gradient of the penalty function using Wirtingerderivatives includes computing a scaled value of field intensity on theimage plane; configuring at least one spatial light modulator using thehologram; and illuminating the spatial light modulator using a lightsource to create the reconstructed image.
 2. The method of claim 1wherein modeling the hologram using a forward propagation model includesmodeling the hologram using an angular spectrum, Fourier, Fraunhofer, orFresnel propagation model.
 3. The method of claim 1 wherein modeling thehologram using a forward propagation model includes modeling thehologram as a spatially invariant convolution of the hologram and a lensphase function.
 4. The method of claim 3 wherein the optimizationproblem uses a least squares optimization function.
 5. The method ofclaim 4 wherein the least squares optimization function comprises:${\underset{H}{minimize}\mspace{14mu}{{{{H*L}}^{2} - I}}_{2}^{2}},$where H is the hologram, L is the lens phase function, and I is thesquare of a target image intensity.
 6. The method of claim 5 wherein thehologram comprises a phase hologram for a phase only spatial lightmodulator and the function$\underset{H}{minimize}\mspace{14mu}{{{{H*L}}^{2} - I}}_{2}^{2}$ isapproximated using the function:${\underset{\phi}{\arg\;\min}\mspace{14mu}{{{{H*L}}^{2} - I}}_{2}^{2}},$where Φ is the phase angle of the hologram for a given object point. 7.The method of claim 6 wherein the function$\underset{\phi}{\arg\;\min}\mspace{14mu}{{{{H*L}}^{2} - I}}_{2}^{2}$is approximated using the following function:${\underset{\phi}{\arg\;\min}\mspace{14mu}{{{{{F^{H}({FH})}({FL})}}^{2} - I}}_{2}^{2}},$where F is the Fourier transform operator and F^(H) is the inverseFourier transform operator.
 8. The method of claim 7 wherein computingthe hologram includes calculating a gradient of the function$\underset{\phi}{\arg\;\min}\mspace{14mu}{{{{{{F^{H}({FH})}({FL})}}^{2} - I}}_{2}^{2}.}$9. The method of claim 1 wherein configuring at least one spatial lightmodulator includes configuring a single phase only spatial lightmodulator.
 10. The method of claim 1 wherein configuring at least onespatial light modulator includes configuring each of a pair of cascadedphase only spatial light modulators.
 11. The method of claim 1 whereincomputing the hologram includes using the penalty function to quantifyerror between the reconstructed image produced by a hologram H and atarget image.
 12. The method of claim 11 wherein the penalty function isbased on perceptual accuracy or a learned perceptual similarity metric.13. The method of claim 11 wherein the hologram comprises a phase onlyhologram and computing the hologram as a solution to the optimizationproblem comprises determining a phase angle of the hologram H thatminimizes the error.
 14. A system for digital holography, the systemcomprising: a display device comprising: at least one processor; ahologram calculator implemented by the at least one processor andconfigured for modeling a hologram using a forward propagation modelthat models propagation of a light field from a hologram plane to animage plane and computing the hologram as a solution to an optimizationproblem that is based on the model, wherein computing the hologram as asolution to an optimization problem includes evaluating holograms aspotential solutions to the optimization problem by computing anapproximation of a gradient of a penalty function with respect to acomplex wave field of a reconstructed image on the image plane usingWirtinger derivatives, wherein the penalty function quantifies adifference between the amplitude of the complex wave field of thereconstructed image on the image plane and an amplitude of a complexwave field of the target image on the image plane, adjusting a phase ofa spatial light modulator based on the penalty function, whereincomputing an approximation of the gradient of the penalty function usingWirtinger derivatives includes computing a scaled value of fieldintensity on the image plane; at least one spatial light modulator; aspatial light modulator controller for configuring the at least onespatial light modulator using the hologram; and a light source forilluminating the at least one spatial light modulator to create thereconstructed image.
 15. The system of claim 14 wherein modeling thehologram using a forward propagation model includes modeling thehologram using an angular spectrum, Fourier, Fraunhofer, or Fresnelpropagation model.
 16. The system of claim 14 wherein modeling thehologram using a forward propagation model includes modeling thehologram as a spatially invariant convolution of the hologram and a lensphase function.
 17. The system of claim 16 wherein the optimizationproblem uses a least squares optimization function.
 18. The system ofclaim 17 wherein the least squares optimization function comprises:$\underset{H}{minimize}\mspace{14mu}{{{{H*L}}^{2} - I}}_{2}^{2}$where H is the hologram, L is the lens phase function, and r is thesquare of a target image intensity.
 19. The system of claim 18 whereinthe hologram comprises a phase hologram for a phase only spatial lightmodulator and the function$\underset{H}{minimize}\mspace{14mu}{{{{H*L}}^{2} - I}}_{2}^{2}$ isapproximated using the function:${\underset{\phi}{\arg\;\min}\mspace{14mu}{{{{H*L}}^{2} - I}}_{2}^{2}},$where Φ is the phase angle of the hologram for a given object point. 20.The system of claim 19 wherein the function$\underset{\phi}{\arg\;\min}\mspace{14mu}{{{{H*L}}^{2} - I}}_{2}^{2}$is approximated using the following function:${\underset{\phi}{\arg\;\min}\mspace{14mu}{{{{{F^{H}({FH})}({FL})}}^{2} - I}}_{2}^{2}},$where F is the Fourier transform operator and F^(H) is the inverseFourier transform operator.
 21. The system of claim 20 wherein computingthe hologram includes calculating a gradient of the function$\underset{\phi}{\arg\;\min}\mspace{14mu}{{{{{{F^{H}({FH})}({FL})}}^{2} - I}}_{2}^{2}.}$22. The system of claim 14 wherein computing the hologram includes usingthe penalty function to quantify error between the reconstructed imageproduced by a hologram H and a target image.
 23. The system of claim 22wherein the hologram comprises a phase only hologram and computing thehologram as a solution to the optimization problem comprises determininga phase angle of the hologram H that minimizes the error.
 24. The systemof claim 22 wherein the penalty function is based on perceptual accuracyor a learned perceptual similarity metric.
 25. The system of claim 14wherein the at least one spatial light modulator comprises a singlephase only spatial light modulator.
 26. The system of claim 14 whereinthe at least one spatial light modulator comprises a pair of cascadedphase only spatial light modulators.
 27. A non-transitory computerreadable medium having stored thereon executable instructions that whenexecuted by a processor of a computer control the computer to performsteps comprising: modeling a hologram using a forward propagation modelthat models propagation of a light field from a hologram plane to animage plane; computing the hologram as a solution to an optimizationproblem that is based on the model, wherein computing the hologram as asolution to an optimization problem includes evaluating holograms aspotential solutions to the optimization problem by computing anapproximation of a gradient of a penalty function with respect to acomplex wave field of a reconstructed image on the image plane usingWirtinger derivatives, wherein the penalty function quantifies adifference between the amplitude of the complex wave field of thereconstructed image on the image plane and an amplitude of a complexwave field of the target image on the image plane, adjusting a phase ofa spatial light modulator based on the penalty function, whereincomputing an approximation of the gradient of the penalty function usingWirtinger derivatives computing a scaled value of field intensity on theimage plane; configuring at least one spatial light modulator using thehologram; and illuminating the spatial light modulator using a lightsource to create the reconstructed image.